Let $f$ and $g$ be bilinear forms on a finite dimensional vector space $V$. Suppose $g$ is non-degenerate. Show that there exists unique linear operator $S$ and $T$ on $V$ such that $$ f(a,b)=g(Sa,b)=g(a,Tb) $$ for all $a,b \in V.$
What happens if $g$ is degenerate?